\begin{frame}
\frametitle{Localiser l'activité}

\centering
\input{vector/active-equation.tikz}
\begin{columns}
  \column{.65\textwidth}\centering
  \input{vector/active-quiescent-reprensentation.tikz}
  \column{.35\textwidth}
\end{columns}
\end{frame}

\begin{frame}[label=this one]
  \frametitle{Simulations \mgs et sous-collection active}
  \begin{itemize}
    \item Trajectoires dans \mgs
      \begin{center}
        \begin{tikzpicture} [scale=0.4,baseline,
          cell/.style={minimum size=0.35cm}]
          \fill (0.65,0.65) rectangle (8.35,8.35);
          \foreach \p/\c in%
          { { (1,8)/forest},{(2,8)/forest},{(3,8)/forest},{(4,8)/forest},{(5,8)/forest},{(6,8)/forest},{(7,8)/forest},{(8,8)/forest}%
            ,{(1,7)/forest},{(2,7)/forest},{(3,7)/forest},{(4,7)/forest},{(5,7)/forest},{(6,7)/forest},{(7,7)/forest},{(8,7)/forest}%
            ,{(1,6)/forest},{(2,6)/forest},{(3,6)/fire},  {(4,6)/fire},  {(5,6)/forest},{(6,6)/forest},{(7,6)/forest},{(8,6)/forest}%
            ,{(1,5)/forest},{(2,5)/forest},{(3,5)/forest},{(4,5)/fire},  {(5,5)/forest},{(6,5)/forest},{(7,5)/forest},{(8,5)/forest}%
            ,{(1,4)/forest},{(2,4)/forest},{(3,4)/forest},{(4,4)/forest},{(5,4)/forest},{(6,4)/forest},{(7,4)/forest},{(8,4)/forest}%
            ,{(1,3)/forest},{(2,3)/forest},{(3,3)/forest},{(4,3)/forest},{(5,3)/forest},{(6,3)/forest},{(7,3)/forest},{(8,3)/forest}%
            ,{(1,2)/forest},{(2,2)/forest},{(3,2)/forest},{(4,2)/forest},{(5,2)/forest},{(6,2)/forest},{(7,2)/forest},{(8,2)/fire}%
            ,{(1,1)/forest},{(2,1)/forest},{(3,1)/forest},{(4,1)/forest},{(5,1)/forest},{(6,1)/forest},{(7,1)/forest},{(8,1)/fire}}
          \node[cell,fill=\c,draw=white,line width=1pt] at \p {};
          \node at (4.5,0) {$C^0$};
        \end{tikzpicture}
        \hfill
        \begin{tikzpicture} [scale=0.4,baseline,
          cell/.style={minimum size=0.35cm}]
          \fill (0.65,0.65) rectangle (8.35,8.35);
          \foreach \p/\c in%
          { { (1,8)/forest},{(2,8)/forest},{(3,8)/forest},{(4,8)/forest},{(5,8)/forest},{(6,8)/forest},{(7,8)/forest},{(8,8)/forest}%
            ,{(1,7)/forest},{(2,7)/fire},  {(3,7)/fire},  {(4,7)/fire},  {(5,7)/fire},  {(6,7)/forest},{(7,7)/forest},{(8,7)/forest}%
            ,{(1,6)/forest},{(2,6)/fire},  {(3,6)/ashes}, {(4,6)/ashes}, {(5,6)/fire},  {(6,6)/forest},{(7,6)/forest},{(8,6)/forest}%
            ,{(1,5)/forest},{(2,5)/fire},  {(3,5)/fire},  {(4,5)/ashes}, {(5,5)/fire},  {(6,5)/forest},{(7,5)/forest},{(8,5)/forest}%
            ,{(1,4)/forest},{(2,4)/forest},{(3,4)/fire},  {(4,4)/fire},  {(5,4)/fire},  {(6,4)/forest},{(7,4)/forest},{(8,4)/forest}%
            ,{(1,3)/forest},{(2,3)/forest},{(3,3)/forest},{(4,3)/forest},{(5,3)/forest},{(6,3)/forest},{(7,3)/fire},  {(8,3)/fire}%
            ,{(1,2)/forest},{(2,2)/forest},{(3,2)/forest},{(4,2)/forest},{(5,2)/forest},{(6,2)/forest},{(7,2)/fire},  {(8,2)/ashes}%
            ,{(1,1)/forest},{(2,1)/forest},{(3,1)/forest},{(4,1)/forest},{(5,1)/forest},{(6,1)/forest},{(7,1)/fire},  {(8,1)/ashes}}
          \node[cell,fill=\c,draw=white,line width=1pt] at \p {};
          \node at (4.5,0) {$C^1 = T(C^0)$};
        \end{tikzpicture}
        \hfill
        \begin{tikzpicture} [scale=0.4,baseline,
          cell/.style={minimum size=0.35cm}]
          \fill (0.65,0.65) rectangle (8.35,8.35);
          \foreach \p/\c in%
          { { (1,8)/fire},  {(2,8)/fire},  {(3,8)/fire},  {(4,8)/fire},  {(5,8)/fire},  {(6,8)/fire},  {(7,8)/forest},{(8,8)/forest}%
            ,{(1,7)/fire},  {(2,7)/ashes}, {(3,7)/ashes}, {(4,7)/ashes}, {(5,7)/ashes}, {(6,7)/fire},  {(7,7)/forest},{(8,7)/forest}%
            ,{(1,6)/fire},  {(2,6)/ashes}, {(3,6)/ashes}, {(4,6)/ashes}, {(5,6)/ashes}, {(6,6)/fire},  {(7,6)/forest},{(8,6)/forest}%
            ,{(1,5)/fire},  {(2,5)/ashes}, {(3,5)/ashes}, {(4,5)/ashes}, {(5,5)/ashes}, {(6,5)/fire},  {(7,5)/forest},{(8,5)/forest}%
            ,{(1,4)/fire},  {(2,4)/fire},  {(3,4)/ashes}, {(4,4)/ashes}, {(5,4)/ashes}, {(6,4)/fire},  {(7,4)/fire},  {(8,4)/fire}%
            ,{(1,3)/forest},{(2,3)/fire},  {(3,3)/fire},  {(4,3)/fire},  {(5,3)/fire},  {(6,3)/fire},  {(7,3)/ashes}, {(8,3)/ashes}%
            ,{(1,2)/forest},{(2,2)/forest},{(3,2)/forest},{(4,2)/forest},{(5,2)/forest},{(6,2)/fire},  {(7,2)/ashes}, {(8,2)/ashes}%
            ,{(1,1)/forest},{(2,1)/forest},{(3,1)/forest},{(4,1)/forest},{(5,1)/forest},{(6,1)/fire},  {(7,1)/ashes}, {(8,1)/ashes}}
          \node[cell,fill=\c,draw=white,line width=1pt] at \p {};
          \node at (4.5,0) {$C^2 = T(C^1) = T^2(C^0)$};
        \end{tikzpicture}

      \end{center}
    \item Décomposition de l'activité et trajectoires
      \begin{itemize}
        \item \textcolor{active}{Sous-collection active $A_i$} et
          \textcolor{quiescent}{sous-collection quiescente $Q_i$}
        \item Décomposition de la relation d'évolution
          \[ A^{i+1} + Q^{i+1} = T(A^i + Q^i) \]
      \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Relation entre $A^{i+1},Q^{i+1}$ et $A^i,Q^i$}
  $$ A^{i+1} + Q^{i+1} = T(A^i + Q^i) $$
  \begin{itemize}
    \item Sous-collection \textcolor{frontier}{frontière $F^i$}
  \end{itemize}
  \begin{columns}
    \column{.5\textwidth}
    \begin{enumerate}
      \item \structure{Propriété} $T(A^i + Q^i) = T(A^i \mid Q^i) + Q^i$
      \item \structure{Propriété} $| A^{i+1} | \subseteq | A^i + F^i |$
      \item \structure{Propriété} $| Q^i - F^i | \subseteq | Q^{i+1} |$
      \item \structure{Définition} $F^i = \text{Lk}(A^i)$
      \item \structure{Remarque} $T(A^i \mid Q^i) = T(A^i \mid F^i)$
      \item \structure{Résultat} $C^{i+1} = T(A^i \mid F^i) + F^i + (Q^i - F^i)$
    \end{enumerate}
    \column{.5\textwidth}
    $$ \text{Lk} \left(
      \tikz[baseline] \node {\includegraphics[width=2cm]{vector/operateursS}};
    \right) =
      \tikz[baseline] \node {\includegraphics[width=2cm]{vector/operateursLkS}};
    $$
  \end{columns}

  \bigskip
      \[
        \left\{
          \arraycolsep=1.4pt%\def\arraystretch{2.2}
          \begin{array}{rl}
            A^0   &= \text{PatMatch}_T(C^0)\\
            A^{i+1} &= \text{PatMatch}_T\left[ T(A^i \mid \text{Lk}(A^i)) + \text{Lk}(A^i) \right]
          \end{array}
        \right.
      \]
\end{frame}
